Some greedyt-intersecting families of finite sequences

Abstract

Letn, s1,s2, ... ands n be positive integers. Assume\(\mathcal{M}(s_1 ,s_2 , \cdots ,s_n ) = \{ (x_1 ,x_2 , \cdots x_n )|0 \leqslant x_i \leqslant s_i ,x_i\) is an integer for eachi}. For\(a = (a_1 ,a_2 , \cdots a_n ) \in \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )\),\(\mathcal{F} \subseteq \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )\), and\(A \subseteq \{ 1,2, \cdots ,n\}\), denotes p (a)={j|1≤j≤n,a j ≥p},\(S_p (\mathcal{F}) = \{ s_p (a)|a \in \mathcal{F}\}\), and\(W_p (A) = p^{n - |A|} \prod\limits_{i \in A} {(s_i - p)}\).\(\mathcal{F}\) is called anI t p -intersecting family if, for any a,b∈\(\mathcal{F}\),a i Λb i =min(a i ,b i )≥p for at leastt i's.\(\mathcal{F}\) is called a greedyI t P -intersecting family if\(\mathcal{F}\) is anI t p -intersecting family andW p (A)≥W p (B+A c ) for anyAeS p (\(\mathcal{F}\)) and any\(B \subseteq A\) with |B|=t−1.